Solve by completing the square.

p2 + 6p = –1

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.


To solve the quadratic equation \( p^2 + 6p = -1 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side
We need to isolate the quadratic and linear terms on the left side of the equation:
\[
p^2 + 6p = -1.
\]
Move the constant on the right side:
\[
p^2 + 6p + \_ = -1 + \_.
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \( p \), square it, and add it to both sides of the equation. The coefficient of \( p \) is 6, so:
\[
\left( \frac{6}{2} \right)^2 = 3^2 = 9.
\]
Add 9 to both sides:
\[
p^2 + 6p + 9 = -1 + 9,
\]
which simplifies to:
\[
p^2 + 6p + 9 = 8.
\]

### Step 3: Write the left side as a perfect square
The left-hand side is now a perfect square:
\[
(p + 3)^2 = 8.
\]

### Step 4: Solve for \( p \)
Take the square root of both sides:
\[
p + 3 = \pm \sqrt{8}.
\]
Since \( \sqrt{8} = 2\sqrt{2} \), this becomes:
\[
p + 3 = \pm 2\sqrt{2}.
\]

### Step 5: Isolate \( p \)
Solve for \( p \) by subtracting 3 from both sides:
\[
p = -3 \pm 2\sqrt{2}.
\]

### Step 6: Simplify the solution
We can leave the solution in its exact form as:
\[
p = -3 + 2\sqrt{2} \quad \text{or} \quad p = -3 - 2\sqrt{2}.
\]
To express the solutions as decimals, approximate \( \sqrt{2} \approx 1.41 \):
\[
p = -3 + 2(1.41) = -3 + 2.82 = -0.18,
\]
\[
p = -3 - 2(1.41) = -3 - 2.82 = -5.82.
\]

### Final answers:
\[
p \approx -0.18 \quad \text{or} \quad p \approx -5.82.
\]

Let me know if you'd like more details on any steps!

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Here are 5 follow-up questions related to solving quadratic equations by completing the square:
1. How do you determine the number to complete the square?
2. Why do we add the same number to both sides when completing the square?
3. What is the geometric interpretation of completing the square?
4. How would you solve \( p^2 - 4p = 5 \) by completing the square?
5. How does the quadratic formula relate to completing the square?

**Tip:** Completing the square is useful for solving quadratics, deriving the quadratic formula, and finding the vertex of a parabola!

Solve by completing the square.

p^2 + 6p = -1

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

Learn how to solve the quadratic equation p^2 + 6p = -1 by completing the square. This step-by-step solution covers key algebraic concepts, including completing the square and solving quadratic equations, with answers provided in decimal form. Ideal for students in Grades 9-11.

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